L(s) = 1 | − 2·2-s − 5.24·3-s + 4·4-s + 5·5-s + 10.4·6-s − 8·8-s + 0.485·9-s − 10·10-s − 1.68·11-s − 20.9·12-s + 38.2·13-s − 26.2·15-s + 16·16-s − 68.9·17-s − 0.970·18-s − 40.6·19-s + 20·20-s + 3.37·22-s + 120.·23-s + 41.9·24-s + 25·25-s − 76.5·26-s + 139.·27-s + 78.8·29-s + 52.4·30-s − 229.·31-s − 32·32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.00·3-s + 0.5·4-s + 0.447·5-s + 0.713·6-s − 0.353·8-s + 0.0179·9-s − 0.316·10-s − 0.0462·11-s − 0.504·12-s + 0.816·13-s − 0.451·15-s + 0.250·16-s − 0.983·17-s − 0.0127·18-s − 0.490·19-s + 0.223·20-s + 0.0326·22-s + 1.09·23-s + 0.356·24-s + 0.200·25-s − 0.577·26-s + 0.990·27-s + 0.504·29-s + 0.319·30-s − 1.33·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 5.24T + 27T^{2} \) |
| 11 | \( 1 + 1.68T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 68.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 40.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 78.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 229.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 344.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 139.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 224.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 37.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 217.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 207.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 92.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 873.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 495.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 362.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 737.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.27847497961099059308136024120, −9.144159992658927858809132564426, −8.537352906114624963476751133937, −7.22174967099541982022648114048, −6.33336961916090019116034444198, −5.67541778701140099340676040458, −4.46839766663131490311202300412, −2.80662065404681690344670275824, −1.33921706017389135457801376258, 0,
1.33921706017389135457801376258, 2.80662065404681690344670275824, 4.46839766663131490311202300412, 5.67541778701140099340676040458, 6.33336961916090019116034444198, 7.22174967099541982022648114048, 8.537352906114624963476751133937, 9.144159992658927858809132564426, 10.27847497961099059308136024120